'Approximating' triangles by ellipses

Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.

Question: Given a general triangle T, to find the ellipse E that minimizes the Hausdorff distance from T. It is not even clear to me if E is unique.

Remarks: One can ask how this ellipse relates to the Steiner ellipses of T, whether the area/perimeter of E(assuming it is unique) is related to area/perimeter of T and so on..

Further questions: if E is somehow found, then is T that triangle (or one of the triangles) that minimizes Hausdorff distance from E? Is there a mapping from ellipses to triangles using this closeness criterion - ie. given any ellipse, is it the closest ellipse to some triangle(s)? This same question can be asked the other way around as well.

Another question: to find closest triangle to a given convex n-gon. And (not) finally, one can replace triangles with quadrilaterals/rectangles and so on!
Guess: If one uses the Frechet distance rather than Hausdorff to formulate the above question, the answer(s) are not affected.
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Update (Jan 18th, 2023): The mathoverflow page for this question is here . It has been pointed out there that the special case of approximating a triangle with a circle is itself of non-trivial interest.

Update (Feb 5th 2023): A further mathoverflow page on the same theme is here